介质密度反演偏导矩阵的精确计算方法
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摘要
实现反演偏导矩阵的计算是基于导数最优化反演方法的关键,然而目前的地震反演几乎都是基于Zoeppritz方程近似实现的,使计算精度和适应范围受到限制.本文利用Zoeppritz方程建立了反射系数对地层介质密度比偏导方程,导出了Zoeppritz方程矩阵元对介质密度比的导数.通过求解偏导方程获得了反射系数对介质密度比偏导数的精确计算(考虑了速度中含介质密度的问题).利用数值算例分析了反射系数对介质密度比偏导数的变化特点.本文采用直接解法求解偏导矩阵方程组,获得了快的计算速度和高的计算精度,为实现地层介质密度反演(包括大角度反演)提供了偏导矩阵的计算方法.
Realization of derivative computation is the key for derivative-based inversion.Its computational precision directly influences the inversion result.However, at present almost all AVO (Amplitude Versus Offset) inversions are based on the approximate expressions of Zoeppritz equations.The computational precision and the application scope of AVO inversions are restricted.With the help of Zoeppritz equations, this paper sets up the partial derivative equation of the reflection coefficients of seismic wave with respect to the ratio of medium densities, and derives the derivative of any matrix element with respect to the ratio of medium densities. Through solving the derivative matrix equation, we have obtained the partial derivative of reflection coefficients of seismic wave with respect to the ratio of medium densities, realized the accurate calculation of derivative matrix equation, which can be used to inverse for the ratio of medium densities.We have also analyzed the derivative curves of reflection coefficients of seismic wave, and achieved some new cognitions for the derivative.Because the method of computational derivative matrix adopted is directly solving the linear system of equations, the computation not only has high precision but also has fast computational speed.This paper offers a computational method for us to further research the inversion of stratum densities (including the inversion problem of large angle incident wave) and to improve the computational speed and precision of inversion.
Realization of derivative computation is the key for derivative-based inversion.Its computational precision directly influences the inversion result.However, at present almost all AVO (Amplitude Versus Offset) inversions are based on the approximate expressions of Zoeppritz equations.The computational precision and the application scope of AVO inversions are restricted.With the help of Zoeppritz equations, this paper sets up the partial derivative equation of the reflection coefficients of seismic wave with respect to the ratio of medium densities, and derives the derivative of any matrix element with respect to the ratio of medium densities. Through solving the derivative matrix equation, we have obtained the partial derivative of reflection coefficients of seismic wave with respect to the ratio of medium densities, realized the accurate calculation of derivative matrix equation, which can be used to inverse for the ratio of medium densities.We have also analyzed the derivative curves of reflection coefficients of seismic wave, and achieved some new cognitions for the derivative.Because the method of computational derivative matrix adopted is directly solving the linear system of equations, the computation not only has high precision but also has fast computational speed.This paper offers a computational method for us to further research the inversion of stratum densities (including the inversion problem of large angle incident wave) and to improve the computational speed and precision of inversion.
引文
[1] Batzle M,Wang Z.Seismic properties of pore fluids.Geophysics,1992,57:1396～1408
[2] Han D H,Nur A,Morgan D.Effects of porosity and clay content on wave velocities in sandstones.Geophysics,1986,51:2093～2107
[3] Keys R G,Xu S Y.An approximation for the Xu-White velocity model.Geophysics,2002,67:1406～1414
[4] Kuster G T,Toks(o|¨)z M N.Velocity and attenuation of seismic waves in two phase media:Part I:Theoretical formulation.Geophysics,1974,39 :587～606
[5] White L,Castagna J.Stochastic fluid modulus inversion.Geophysics,2002,67:1853～1843
[6] Xu S,White R E.A new velocity model for clay-sand mixtures.Geophysical Prospecting,1995,43 :91～118
[7] Fred J H 著.地震振幅解释.孙夕平,赵良武译.北京:石油工业出版社,2006:18～39 Fred J H.Seismic Amplitude Interpretation(in Chinese).Sun X P,Zhao L W trans.Beijing:Petroleum Industry Press,2006:18～39
[8] 陆基孟.地震勘探原理(上,下).山东东营:中国石油大学出版社,1993,上:20～30;下:123～194,267 Lu J M.Seismic Exploration Principle(the First and Second Volumes)(in Chinese).Shandong Dongying:China University of Petroleum Press,1993,the first volume:20～ 30;the second volume:123～194,267
[9] Xu S,White R E.A physical model for shear-wave velocity prediction.Geophysical Prospecting,1996,44:687～717
[10] Shuey R T.A simplification of the Zoeppritz-equations.Geophysics,1985,50(3) :609～614
[11] 马在田.P-SV反射转换波的倾角时差校正(DMO)方法研究.地球物理学报,1996,39(2) :243～250 Ma Z T.DMO for P-SV converted reflection.Chinese J.Geophys.(in Chinese),1996,39(2) :243～250
[12] 李景叶,陈小宏,郝振江等.多波时移地震AVO反演研究.地球物理学报,2005,48(4) :902～908 Li J Y,ChenXH,HaoZJ,et al.A study on multiple timelapse seismic AVO inversion.Chinese J.Geophys.(in Chinese),2005,48(4) :902～908
[13] 孟宪军,姜秀娣,黄捍东等.叠前AVA广义非线性纵横波速度反演.石油地球物理勘探,2004,39(6) :645～650 Meng X J,Jiang X D,Huang H D,et al.Generalized nonlinear P wave and S wave velocity inversion of prestack AVA.Oil Geophysical Prospecting(in Chinese),2004,39(6) :645～650
[14] Shou H,Liu H,Gao J H.AVO inversion based on common shot migration.Applied Geophysics,2006,3(2) :99～104
[15] 顾汉明,王家映,江涛等.海底多波多分量AVO蕴含弹性参数信息的定量分析.石油地球物理勘探,2000,35(1) :56～63 Gu H M,Wang J Y,Jiang T,et al.Quantitative analysis of elastic parameter information contained in AVO of sea-bottom multi-wave multi-component seismic data.Oil Geophysical Prospecting(in Chinese),2000,35(1) :56～63
[16] 陈建江,印兴耀,张广智.层状介质AVO叠前反演.石油地球物理勘探,2006,41(6) :656～662 Chen J J,Yin X Y,Zhang G Z.Prestack AVO inversion of layered medium.Oil Geophysical Prospecting(in Chinese),2006,41(6) .656～662
[17] 王保丽,印兴耀,张繁昌.基于Gray近似的弹性波阻抗方程及反演.石油地球物理勘探,2007,42(4) :435～439 Wang B L,Yin X Y,Zhang F C.Gray approximation-based elastic wave impedance equation and inversion.Oil Geophysical Prospecting(in Chinese),2007,42(4) :435～439
[18] 丁继才,常旭,刘伊克等.基于声波方程的井间地震数据快速WTW反演方法.地球物理学报,2007,50(5) :1527～1533 Ding J C,Chang X,Liu Y K,et al.Rapid method for acoustic wave-equation WTW inversion of cross hole seismic data.Chinese J.Geophys.(in Chinese),2007,50(5) :1527～1533
[19] Brittan J,Warner M.Wide-angle seismic velocities in heterogeneous crust.Geophysics Journal International,1997,129(2) :269～280
[20] Zhang W P,Guo P,Hu T Y.Study and practice of wide angle seismic data processing.Applied Geophysics,2004,1(1) :31～37
[21] Fruehn J,White R S,Fliedner M,et al.Large-aperture seismic:imaging beneath high-velocity strata.World Oil,1999,220(1) :109～113
[22] 刘福平,李瑞忠,杨长春等.非均匀P-偏振电磁波在导电界面的类全反射横向偏移.地球物理学报,2007,50(2) :556～566 Liu F P,Li R Z,Yang C C,et al.The lateral shift of quasi total reflection of inhomogeneous P-electromagnetic wave on the interface of conductive media.Chinese J.Geophys.(in Chinese),2007,50(2) :556～566
[23] Liu F P,Wang A L,Chen Q,et al.The research progress on lateral shift of reflected electromagnetic wave at the interface of two conductive media.Chinese Science Bulletin,2008,53(7) :961～968
[24] 胡中平,管路平,顾连兴等.高速屏蔽层下广角地震波场分析及成像方法.地球物理学报,2004,47(1) :88～94 Hu Z P,Guan L P,Gu L X,et al.Wide angle seismic wave field analysis and imaging method below the high velocity shield layers.Chinese J.Geophys.(in Chinese),2004,47(1) :88～94
[25] 李怀胜.广角反射在塔中地区的应用.石油物探,2005,44(3) :292～295 Li H S.The application of wide-angle reflection method in central Tarim basin.Geophysical Prospecting for Petroleum(in Chinese),2005,44(3) :292～295
[26] 王志,贺振华,黄德济等.高速屏蔽层下弱反射层地震勘探--广角反射.勘探地球物理进展,2002,25(5) :23～27 Wang Z,He Z H,Huang D J,et al.A seismic survey method for weak reflectors below shielded layer of high velocity:wide angle reflection.Progress in Exploration Geophysics(in Chinese),2002,25(5) :23～27
[27] 孙成禹,倪长宽,李胜军等.广角地震反射数据特征及校正方法研究.石油地球物理勘探,2007,42(1) :24～29 Sun C Y,Ni C K,Li S J,et al.Feature of wide-angle reflection data and correction method.Oil Geophysical Prospecting(in Chinese),2007,42(1) :24～29
[28] 王志,贺振华,黄德济等.广角反射波场特征研究及正演模拟分析.地球物理学进展,2003,18(1) :116～121 Wang Z,He Z H,Huang D J,et al.The wave field features researching of wide-angle reflection and the analyzing of forward modeling.Progress in Geophysics(in Chinese),2003,18(1) :116～121
[29] 徐文君,於文辉,胡中平.广角反射波的特征及正演模拟.石油地球物理勘探,2006,41(4) :390～395 Xu W J,Yu W H,Hu Z P.Feature and forward simulation of wide angle reflection.Oil Geophysical Prospecting(in Chinese),2006,41(4) :390～395
[30] 王彦飞.反演问题的计算方法及其应用--当代科学前沿论丛.北京:高等教育出版社,2007:31～84 Wang Y F.Computational Method for Inverse Problem and Their Applications(in Chinese).Beijing:Higher Education Press,2007 :31～84
[31] 刘福平,孟宪军,王玉梅等.反演纵横波速度的Jacobi矩阵及精确计算方法.中国科学D,2010,40(9) :(待发表)Liu F P,Meng X J,Wang Y M,et al.Jacobian matrix for the inversion of P-and S-wave velocities and its accurate computation method.Science in China(Series D),2010,40(9) :(in press)
[2] Han D H,Nur A,Morgan D.Effects of porosity and clay content on wave velocities in sandstones.Geophysics,1986,51:2093～2107
[3] Keys R G,Xu S Y.An approximation for the Xu-White velocity model.Geophysics,2002,67:1406～1414
[4] Kuster G T,Toks(o|¨)z M N.Velocity and attenuation of seismic waves in two phase media:Part I:Theoretical formulation.Geophysics,1974,39 :587～606
[5] White L,Castagna J.Stochastic fluid modulus inversion.Geophysics,2002,67:1853～1843
[6] Xu S,White R E.A new velocity model for clay-sand mixtures.Geophysical Prospecting,1995,43 :91～118
[7] Fred J H 著.地震振幅解释.孙夕平,赵良武译.北京:石油工业出版社,2006:18～39 Fred J H.Seismic Amplitude Interpretation(in Chinese).Sun X P,Zhao L W trans.Beijing:Petroleum Industry Press,2006:18～39
[8] 陆基孟.地震勘探原理(上,下).山东东营:中国石油大学出版社,1993,上:20～30;下:123～194,267 Lu J M.Seismic Exploration Principle(the First and Second Volumes)(in Chinese).Shandong Dongying:China University of Petroleum Press,1993,the first volume:20～ 30;the second volume:123～194,267
[9] Xu S,White R E.A physical model for shear-wave velocity prediction.Geophysical Prospecting,1996,44:687～717
[10] Shuey R T.A simplification of the Zoeppritz-equations.Geophysics,1985,50(3) :609～614
[11] 马在田.P-SV反射转换波的倾角时差校正(DMO)方法研究.地球物理学报,1996,39(2) :243～250 Ma Z T.DMO for P-SV converted reflection.Chinese J.Geophys.(in Chinese),1996,39(2) :243～250
[12] 李景叶,陈小宏,郝振江等.多波时移地震AVO反演研究.地球物理学报,2005,48(4) :902～908 Li J Y,ChenXH,HaoZJ,et al.A study on multiple timelapse seismic AVO inversion.Chinese J.Geophys.(in Chinese),2005,48(4) :902～908
[13] 孟宪军,姜秀娣,黄捍东等.叠前AVA广义非线性纵横波速度反演.石油地球物理勘探,2004,39(6) :645～650 Meng X J,Jiang X D,Huang H D,et al.Generalized nonlinear P wave and S wave velocity inversion of prestack AVA.Oil Geophysical Prospecting(in Chinese),2004,39(6) :645～650
[14] Shou H,Liu H,Gao J H.AVO inversion based on common shot migration.Applied Geophysics,2006,3(2) :99～104
[15] 顾汉明,王家映,江涛等.海底多波多分量AVO蕴含弹性参数信息的定量分析.石油地球物理勘探,2000,35(1) :56～63 Gu H M,Wang J Y,Jiang T,et al.Quantitative analysis of elastic parameter information contained in AVO of sea-bottom multi-wave multi-component seismic data.Oil Geophysical Prospecting(in Chinese),2000,35(1) :56～63
[16] 陈建江,印兴耀,张广智.层状介质AVO叠前反演.石油地球物理勘探,2006,41(6) :656～662 Chen J J,Yin X Y,Zhang G Z.Prestack AVO inversion of layered medium.Oil Geophysical Prospecting(in Chinese),2006,41(6) .656～662
[17] 王保丽,印兴耀,张繁昌.基于Gray近似的弹性波阻抗方程及反演.石油地球物理勘探,2007,42(4) :435～439 Wang B L,Yin X Y,Zhang F C.Gray approximation-based elastic wave impedance equation and inversion.Oil Geophysical Prospecting(in Chinese),2007,42(4) :435～439
[18] 丁继才,常旭,刘伊克等.基于声波方程的井间地震数据快速WTW反演方法.地球物理学报,2007,50(5) :1527～1533 Ding J C,Chang X,Liu Y K,et al.Rapid method for acoustic wave-equation WTW inversion of cross hole seismic data.Chinese J.Geophys.(in Chinese),2007,50(5) :1527～1533
[19] Brittan J,Warner M.Wide-angle seismic velocities in heterogeneous crust.Geophysics Journal International,1997,129(2) :269～280
[20] Zhang W P,Guo P,Hu T Y.Study and practice of wide angle seismic data processing.Applied Geophysics,2004,1(1) :31～37
[21] Fruehn J,White R S,Fliedner M,et al.Large-aperture seismic:imaging beneath high-velocity strata.World Oil,1999,220(1) :109～113
[22] 刘福平,李瑞忠,杨长春等.非均匀P-偏振电磁波在导电界面的类全反射横向偏移.地球物理学报,2007,50(2) :556～566 Liu F P,Li R Z,Yang C C,et al.The lateral shift of quasi total reflection of inhomogeneous P-electromagnetic wave on the interface of conductive media.Chinese J.Geophys.(in Chinese),2007,50(2) :556～566
[23] Liu F P,Wang A L,Chen Q,et al.The research progress on lateral shift of reflected electromagnetic wave at the interface of two conductive media.Chinese Science Bulletin,2008,53(7) :961～968
[24] 胡中平,管路平,顾连兴等.高速屏蔽层下广角地震波场分析及成像方法.地球物理学报,2004,47(1) :88～94 Hu Z P,Guan L P,Gu L X,et al.Wide angle seismic wave field analysis and imaging method below the high velocity shield layers.Chinese J.Geophys.(in Chinese),2004,47(1) :88～94
[25] 李怀胜.广角反射在塔中地区的应用.石油物探,2005,44(3) :292～295 Li H S.The application of wide-angle reflection method in central Tarim basin.Geophysical Prospecting for Petroleum(in Chinese),2005,44(3) :292～295
[26] 王志,贺振华,黄德济等.高速屏蔽层下弱反射层地震勘探--广角反射.勘探地球物理进展,2002,25(5) :23～27 Wang Z,He Z H,Huang D J,et al.A seismic survey method for weak reflectors below shielded layer of high velocity:wide angle reflection.Progress in Exploration Geophysics(in Chinese),2002,25(5) :23～27
[27] 孙成禹,倪长宽,李胜军等.广角地震反射数据特征及校正方法研究.石油地球物理勘探,2007,42(1) :24～29 Sun C Y,Ni C K,Li S J,et al.Feature of wide-angle reflection data and correction method.Oil Geophysical Prospecting(in Chinese),2007,42(1) :24～29
[28] 王志,贺振华,黄德济等.广角反射波场特征研究及正演模拟分析.地球物理学进展,2003,18(1) :116～121 Wang Z,He Z H,Huang D J,et al.The wave field features researching of wide-angle reflection and the analyzing of forward modeling.Progress in Geophysics(in Chinese),2003,18(1) :116～121
[29] 徐文君,於文辉,胡中平.广角反射波的特征及正演模拟.石油地球物理勘探,2006,41(4) :390～395 Xu W J,Yu W H,Hu Z P.Feature and forward simulation of wide angle reflection.Oil Geophysical Prospecting(in Chinese),2006,41(4) :390～395
[30] 王彦飞.反演问题的计算方法及其应用--当代科学前沿论丛.北京:高等教育出版社,2007:31～84 Wang Y F.Computational Method for Inverse Problem and Their Applications(in Chinese).Beijing:Higher Education Press,2007 :31～84
[31] 刘福平,孟宪军,王玉梅等.反演纵横波速度的Jacobi矩阵及精确计算方法.中国科学D,2010,40(9) :(待发表)Liu F P,Meng X J,Wang Y M,et al.Jacobian matrix for the inversion of P-and S-wave velocities and its accurate computation method.Science in China(Series D),2010,40(9) :(in press)
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